Abstract: A natural class of singular varieties consists of the quotients of vector spaces by linear actions of a finite group G. We can ask what the Chow group of algebraic cycles is, for such a quotient. By taking larger and larger representations of G, we can package these Chow groups into a ring, called the Chow ring of the classifying space of G, or (for short) the Chow ring of G. It maps to the cohomology ring of G, usually not by an isomorphism.

We present the latest tools for computing Chow rings of finite groups. These tools give complete calculations for all “small” groups and many other finite groups. A surprising point is that Chow rings become “wild,” in a precise sense, for some slightly larger finite groups.